This is a question on chap. 9 of Introduction to Set Theory of Hrbacek and Jech. The book define that the Zorn's Lemma is: If every chain in a partially ordered set has an upper bound, then the partially ordered set has a maximal element.
This is very confused to me. I tried find a upper bound for each chain (is the $\subseteq$-maximal element ?) but I don't know how to try this. Furthermore how I will find the maximal element for the partially ordered set?
On
You apply Axiom of Choice every time to pick a larger element for your incomplete chain. By every time, you are constructing a function on all ordinal numbers to your poset, since the function is increasing, so 1-1, eventually it stops at some ordinal, then the corresponding one is your maximal element.
The point of the lemma is that it guarantees the existence of a maximal element. It doesn't tell you which element it is.
Just like the axiom of choice guarantees that there is a function $f\colon\mathcal P(\Bbb R)\to\Bbb R$ such that for every non-empty set $A$, $f(A)\in A$. It doesn't tell us what is $f(\Bbb R)$ or what is $f(\Bbb{R\setminus Q})$ or anything like that. It exists, move along.
Zorn's lemma, the axiom of choice, the well-ordering principle, and so on, only claim that certain objects exist. Since modern mathematics is often non-constructive, we don't have to actually define and construct the objects we use. It suffices to prove their existence, and Zorn's lemma is a wonderful tool for just that.
So Zorn's lemma tells us, essentially, that if whenever we pick a chain in $P$, then we can find an element which is larger (or equal) to all the elements in the chain; then we can be sure that there is a maximal element in $P$.
This upper bound doesn't have to be unique, or "tight" either. It just has to be an upper bound.